New critical point theorem and infinitely many small-magnitude solutions of a nonlinear Iwatsuka model

Abstract

In this study, the following nonlinear magnetic Schrödinger equation is considered−ΔAu+V(x)u=Eu+|u|p−2u,x=(x1,x2)∈R2, where 2<p<+∞, A(x)=(0,−b(x1)), b(x1)=∫0x1B(t)dt, B∈C∞(R,R) is an increasing function with different limits at +∞ and −∞. Under these assumptions, the magnetic Laplacian operator −ΔA is known as an Iwatsuka model, and its spectrum has a band structure. In addition, E is an edge of a spectral gap of operator −ΔA, and V is a power-like decay potential. It is proved that this equation has a sequence of non-zero solutions, whose L∞ norms tend to zero along this sequence. The proof is given by establishing a new critical point theorem without the usual (PS)⁎ condition. This study is a subsequent work to a recent one (Chen, 2022 [11]).